Can't speak from personal experience of dealing with parents but educate yourself with info about ethics, environment, nutrition etc so you can defend yourself from any criticisms. In particular, be aware of the studies that show advantages to veganism on cancer, longevity and heart disease.
Offer to learn to cook a few vegan meals for the family.
Tell the family this is very important for you. Suggest that they watch a movie about veganism with you to understand it.
Expect resistance/uncertainty, learn to deal with it steadily over time by working on people that are not initially accepting.
Perservere.
Good luck.